I think I did not clearly explain the problem I am working on. I am working to better understand time dilation.

One of the illustrations I found is the light clock. While it does mathematically explain the concept of time dilation, the photons of light observed in the green line (D) cannot be the same as the photons observed in the dotted line (L), since the direction of light is not affected by the motion of the light source.

IE, either the light was projected in direction L, or direction D. In the case of a moving target, depending on its velocity, we would see a left curving line L. Can you help me understand this diagram better?

My idea was to plot the motion of a hydrogen atom parallel to the path of the photon which is why I asked about the mathematical expression of the hydrogen atom.

I think I did not clearly explain the problem I am working on. I am working to better understand time dilation.

One of the illustrations I found is the light clock. While it does mathematically explain the concept of time dilation, the photons of light observed in the green line (D) cannot be the same as the photons observed in the dotted line (L), since the direction of light is not affected by the motion of the light source.

IE, either the light was projected in direction L, or direction D. In the case of a moving target, depending on its velocity, we would see a left curving line L. Can you help me understand this diagram better?

My idea was to plot the motion of a hydrogen atom parallel to the path of the photon which is why I asked about the mathematical expression of the hydrogen atom.

The green dotted line L in the second diagram is somewhat confusing. The diagram on the left is the path of the light to one observer (the one moving with the clock). The diagram on the right is the path of the light to a second observer, for whom the clock is moving to the right.

How you got from there to the model of the hydrogen atom is indeed a crooked path!

Here's what a light clock would actually do. So I think the diagram above is incorrect.

Think about tossing a coin on an aeroplane. Does the coin fly backwards and hit the rear of the cabin? Or, does the coin behave normally to someone on the plane?

What path does the coin take to an observer on the ground?

If the coin is in the air for 0.5s, then by my calculations the plane and the coin will have travelled about 150m horizontally while it was tossed. Yet, both observers see the coin leave and return to the passenger's hand.

In this respect, a moving light clock would be no different. Both observers would see the light bounce between the top and bottom. But, one observer would see the whole apparatus move horizontally while this was happening.

The critical difference is that the coin can (and does) have a different velocity in the two frames. But, the invariance of the speed of light means that light cannot move faster to one observer than the other. And this leads to a serious fresh analysis of the concepts of classical physics, including the universality of time itself.

Think about tossing a coin on an aeroplane. Does the coin fly backwards and hit the rear of the cabin? Or, does the coin behave normally to someone on the plane?

What path does the coin take to an observer on the ground?

If the coin is in the air for 0.5s, then by my calculations the plane and the coin will have travelled about 150m horizontally while it was tossed. Yet, both observers see the coin leave and return to the passenger's hand.

In this respect, a moving light clock would be no different. Both observers would see the light bounce between the top and bottom. But, one observer would see the whole apparatus move horiziontally while this was happening.

The critical difference is that the coin can (and does) have a different velocity in the two frames. But, the invariance of the speed of light means that light cannot move faster to one observer than the other. And this leads to a serious fresh analysis of the concepts of classical physics, including the universality of time itself.

Thanks, yes I agree completely regarding the path of the coin, totally get that.

I was going to use a hydrogen atom in place of the coin for another reason.

However, here is what I am getting at. While the path of the coin does indeed change with the motion of the aircraft, a photon in the same situation does not. A photon emitted within the confines of an aircraft would deflect (even though very slightly) perpendicular to the path of the aircraft. This would hold true even at low velocities.
So, I am not seeing how the light clock thought experiment could be correct. The cesium clocks they use for these experiments depend on the rate of decay of the cesium to measure time, not the motion of a photon between to points.

Thanks, yes I agree completely regarding the path of the coin, totally get that.

I was going to use a hydrogen atom in place of the coin for another reason.

However, here is what I am getting at. While the path of the coin does indeed change with the motion of the aircraft, a photon in the same situation does not. A photon emitted within the confines of an aircraft would deflect (even though very slightly) perpendicular to the path of the aircraft. This would hold true even at low velocities.
So, I am not seeing how the light clock thought experiment could be correct. The cesium clocks they use for these experiments depend on the rate of decay of the cesium to measure time, not the motion of a photon between to points.

Your missing the point:

I hope you are not one of these people who think that the velocity of light is independent of the source? It's the speed of light that is invariant. The direction can't be invariant.

The light clock is taken as a example clock. This shows time dilation. It is then inferred that the issue is with time itself - not with any specific clock and that time dilation applies however time is measured.

Adding the coin to the experiment (which I had planned on adding an H atom, but the coin will do).
The coin as you pointed out keeps its relative position with the stationary observer, but the photon deflects.
This is why the light clock thought experiment in the other diagram is incorrect. It gives the light photon a bit of a 'homing mechanism' in that it deflects to follow the intended target.

The light clock diagram does not apply to time dilation, in my opinion, as it is based on incorrect assumptions.

I hope you are not one of these people who think that the velocity of light is independent of the source? It's the speed of light that is invariant. The direction can't be invariant.

The light clock is taken as a example clock. This shows time dilation. It is then inferred that the issue is with time itself - not with any specific clock and that time dilation applies however time is measured.

I think you have missed the point. Are you telling me that a projected light beam will always hit a target along its original line of motion, even if the target moves? What if there are two targets? One moving and one not? Will it hit both?

Adding the coin to the experiment (which I had planned on adding an H atom, but the coin will do).
The coin as you pointed out keeps its relative position with the stationary observer, but the photon deflects.
This is why the light clock thought experiment in the other diagram is incorrect. It gives the light photon a bit of a 'homing mechanism' in that it deflects to follow the intended target.

The light clock diagram does not apply to time dilation, in my opinion, as it is based on incorrect assumptions.

There is no homing mechanism. "Straight up" is frame dependent. What is straight up for the passenger on the plane is not straight up to someone on the ground. The light ray goes "straight up" relative to the clock. That is not "straight up" to someone for whom the clock is moving.

The "moving" clock has no idea that it's moving. It's simply firing a light beam up and down.

There is no homing mechanism. "Straight up" is frame dependent. What is straight up for the passenger on the plane is not straight up to someone on the ground. The light ray goes "straight up" relative to the clock. That is not "straight up" to someone for whom the clock is moving.

The "moving" clock has no idea that it's moving. It's simply firing a light beam up and down.

Yes I know. But please answer one question - will an individual photon emitted at T 0 hit the target at T+15?

If yes, the photon has a homing mechanism.
If no, then the photon has deflected and the light clock thought experiment is invalid.